Properties

Label 21.6.a.c
Level $21$
Weight $6$
Character orbit 21.a
Self dual yes
Analytic conductor $3.368$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [21,6,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 5 q^{2} + 9 q^{3} - 7 q^{4} + 94 q^{5} + 45 q^{6} - 49 q^{7} - 195 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{2} + 9 q^{3} - 7 q^{4} + 94 q^{5} + 45 q^{6} - 49 q^{7} - 195 q^{8} + 81 q^{9} + 470 q^{10} + 52 q^{11} - 63 q^{12} - 770 q^{13} - 245 q^{14} + 846 q^{15} - 751 q^{16} - 2022 q^{17} + 405 q^{18} + 1732 q^{19} - 658 q^{20} - 441 q^{21} + 260 q^{22} - 576 q^{23} - 1755 q^{24} + 5711 q^{25} - 3850 q^{26} + 729 q^{27} + 343 q^{28} + 5518 q^{29} + 4230 q^{30} + 6336 q^{31} + 2485 q^{32} + 468 q^{33} - 10110 q^{34} - 4606 q^{35} - 567 q^{36} - 7338 q^{37} + 8660 q^{38} - 6930 q^{39} - 18330 q^{40} - 3262 q^{41} - 2205 q^{42} + 5420 q^{43} - 364 q^{44} + 7614 q^{45} - 2880 q^{46} + 864 q^{47} - 6759 q^{48} + 2401 q^{49} + 28555 q^{50} - 18198 q^{51} + 5390 q^{52} + 4182 q^{53} + 3645 q^{54} + 4888 q^{55} + 9555 q^{56} + 15588 q^{57} + 27590 q^{58} - 11220 q^{59} - 5922 q^{60} - 45602 q^{61} + 31680 q^{62} - 3969 q^{63} + 36457 q^{64} - 72380 q^{65} + 2340 q^{66} + 1396 q^{67} + 14154 q^{68} - 5184 q^{69} - 23030 q^{70} + 18720 q^{71} - 15795 q^{72} + 46362 q^{73} - 36690 q^{74} + 51399 q^{75} - 12124 q^{76} - 2548 q^{77} - 34650 q^{78} + 97424 q^{79} - 70594 q^{80} + 6561 q^{81} - 16310 q^{82} - 81228 q^{83} + 3087 q^{84} - 190068 q^{85} + 27100 q^{86} + 49662 q^{87} - 10140 q^{88} - 3182 q^{89} + 38070 q^{90} + 37730 q^{91} + 4032 q^{92} + 57024 q^{93} + 4320 q^{94} + 162808 q^{95} + 22365 q^{96} + 4914 q^{97} + 12005 q^{98} + 4212 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
5.00000 9.00000 −7.00000 94.0000 45.0000 −49.0000 −195.000 81.0000 470.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.6.a.c 1
3.b odd 2 1 63.6.a.b 1
4.b odd 2 1 336.6.a.i 1
5.b even 2 1 525.6.a.b 1
5.c odd 4 2 525.6.d.c 2
7.b odd 2 1 147.6.a.f 1
7.c even 3 2 147.6.e.c 2
7.d odd 6 2 147.6.e.d 2
12.b even 2 1 1008.6.a.a 1
21.c even 2 1 441.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.a.c 1 1.a even 1 1 trivial
63.6.a.b 1 3.b odd 2 1
147.6.a.f 1 7.b odd 2 1
147.6.e.c 2 7.c even 3 2
147.6.e.d 2 7.d odd 6 2
336.6.a.i 1 4.b odd 2 1
441.6.a.c 1 21.c even 2 1
525.6.a.b 1 5.b even 2 1
525.6.d.c 2 5.c odd 4 2
1008.6.a.a 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(21))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 5 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 94 \) Copy content Toggle raw display
$7$ \( T + 49 \) Copy content Toggle raw display
$11$ \( T - 52 \) Copy content Toggle raw display
$13$ \( T + 770 \) Copy content Toggle raw display
$17$ \( T + 2022 \) Copy content Toggle raw display
$19$ \( T - 1732 \) Copy content Toggle raw display
$23$ \( T + 576 \) Copy content Toggle raw display
$29$ \( T - 5518 \) Copy content Toggle raw display
$31$ \( T - 6336 \) Copy content Toggle raw display
$37$ \( T + 7338 \) Copy content Toggle raw display
$41$ \( T + 3262 \) Copy content Toggle raw display
$43$ \( T - 5420 \) Copy content Toggle raw display
$47$ \( T - 864 \) Copy content Toggle raw display
$53$ \( T - 4182 \) Copy content Toggle raw display
$59$ \( T + 11220 \) Copy content Toggle raw display
$61$ \( T + 45602 \) Copy content Toggle raw display
$67$ \( T - 1396 \) Copy content Toggle raw display
$71$ \( T - 18720 \) Copy content Toggle raw display
$73$ \( T - 46362 \) Copy content Toggle raw display
$79$ \( T - 97424 \) Copy content Toggle raw display
$83$ \( T + 81228 \) Copy content Toggle raw display
$89$ \( T + 3182 \) Copy content Toggle raw display
$97$ \( T - 4914 \) Copy content Toggle raw display
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